Solve 176x=121+61x^2

$$176 x = 121 + 61 x ^ { 2 }$$

Solve for x

$$x = \frac{11 \sqrt{3} + 88}{61} \approx 1.754959982$$

$$x = \frac{88 - 11 \sqrt{3}}{61} \approx 1.13028592$$

Steps Using the Quadratic Formula

Subtract 121 from both sides.

$$176x-121=61x^{2}$$

Subtract 61x^{2} from both sides.

$$176x-121-61x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-61x^{2}+176x-121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -61 for a, 176 for b, and -121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-176±\sqrt{176^{2}-4\left(-61\right)\left(-121\right)}}{2\left(-61\right)}$$

Square 176.

$$x=\frac{-176±\sqrt{30976-4\left(-61\right)\left(-121\right)}}{2\left(-61\right)}$$

Multiply -4 times -61.

$$x=\frac{-176±\sqrt{30976+244\left(-121\right)}}{2\left(-61\right)}$$

Multiply 244 times -121.

$$x=\frac{-176±\sqrt{30976-29524}}{2\left(-61\right)}$$

Add 30976 to -29524.

$$x=\frac{-176±\sqrt{1452}}{2\left(-61\right)}$$

Take the square root of 1452.

$$x=\frac{-176±22\sqrt{3}}{2\left(-61\right)}$$

Multiply 2 times -61.

$$x=\frac{-176±22\sqrt{3}}{-122}$$

Now solve the equation x=\frac{-176±22\sqrt{3}}{-122} when ± is plus. Add -176 to 22\sqrt{3}.

$$x=\frac{22\sqrt{3}-176}{-122}$$

Divide -176+22\sqrt{3} by -122.

$$x=\frac{88-11\sqrt{3}}{61}$$

Now solve the equation x=\frac{-176±22\sqrt{3}}{-122} when ± is minus. Subtract 22\sqrt{3} from -176.

$$x=\frac{-22\sqrt{3}-176}{-122}$$

Divide -176-22\sqrt{3} by -122.

$$x=\frac{11\sqrt{3}+88}{61}$$

The equation is now solved.

$$x=\frac{88-11\sqrt{3}}{61} x=\frac{11\sqrt{3}+88}{61}$$

Steps for Completing the Square

Subtract 61x^{2} from both sides.

$$176x-61x^{2}=121$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-61x^{2}+176x=121$$

Divide both sides by -61.

$$\frac{-61x^{2}+176x}{-61}=\frac{121}{-61}$$

Dividing by -61 undoes the multiplication by -61.

$$x^{2}+\frac{176}{-61}x=\frac{121}{-61}$$

Divide 176 by -61.

$$x^{2}-\frac{176}{61}x=\frac{121}{-61}$$

Divide 121 by -61.

$$x^{2}-\frac{176}{61}x=-\frac{121}{61}$$

Divide -\frac{176}{61}, the coefficient of the x term, by 2 to get -\frac{88}{61}. Then add the square of -\frac{88}{61} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{176}{61}x+\left(-\frac{88}{61}\right)^{2}=-\frac{121}{61}+\left(-\frac{88}{61}\right)^{2}$$

Square -\frac{88}{61} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{176}{61}x+\frac{7744}{3721}=-\frac{121}{61}+\frac{7744}{3721}$$

Add -\frac{121}{61} to \frac{7744}{3721} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{176}{61}x+\frac{7744}{3721}=\frac{363}{3721}$$

Factor x^{2}-\frac{176}{61}x+\frac{7744}{3721}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{88}{61}\right)^{2}=\frac{363}{3721}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{88}{61}\right)^{2}}=\sqrt{\frac{363}{3721}}$$

Simplify.

$$x-\frac{88}{61}=\frac{11\sqrt{3}}{61} x-\frac{88}{61}=-\frac{11\sqrt{3}}{61}$$

Add \frac{88}{61} to both sides of the equation.

$$x=\frac{11\sqrt{3}+88}{61} x=\frac{88-11\sqrt{3}}{61}$$